Solving transmission expansion planning by
using the traditional model has some drawbacks as follows:
1. The problem is considered deterministically, which is
not appropriate. Market mechanism and uncertainty in the
deregulated environment should be taken into account and the
probabilistic or risk-based approach should be applied.
2. Considering reliability criterion only under the normal
condition is not enough. Inclusion of facility availabilities will
make the problem more realistic.
3. Because many participants are affected by the
transmission expansion, the objective of this problem must be
modified. The expansion that provides the best advantage (socalled
social welfare) for all participants should be selected.
The proposedmethod can be mainly classified in two groups: (i) mathematical optimization algorithms, (ii) heuristic
algorithms. Mathematical optimization method is to formulate the problem of transmission network
using mathematical model, such as linear programming [4, 5] and Benders decomposition approach
[6]. Recently, many heuristic algorithms, etc. simulated annealing (SA) [7], genetic algorithms (GA)
[8, 9], tabu search (TS) [10], ant colony algorithm [11], have been applied to the TNEP problem. These
algorithms are usually robust and generate near-optimal solutions for large complex networks. The
combined use of optimization and heuristics algorithm has also been tried in [12]. The authors in [13]
have investigated the method of SA, GA, TS, and then a hybrid approach based on TS algorithm is
MATHEMATICAL MODEL OF TNEP (Transmission Network Expansion Plan)
The AC power flow equation of power system [1] is:
Where Pi is the input real power flow at bus i, Vi and Vi are the voltages at bus i and j, respectively. Gi j is
the susceptance and Bi j is the conductance. di j is the voltage phase angle between i and j.
Considering the characteristics of high voltage network, the above AC power equation could be
simplified.
1) As the resistance of transmission circuits is significantly less than the reactance, it is
reasonable to approximate G=0.
2) As the voltage phase angle difference is very small, it is reasonable to
approximate sindi j= di j.
3) In the per-unit system, the numerical values of voltage magnitudes are very close to 1. Given the discussed
practical approximations, the power flow in the transmission system can be approximated to the following equation.
Analysis and computation of the DC power flow model are very convenient. The calculated amount is
much less, and the speed could accelerate much more. In the TNEP problem, as the raw data itself are notaccurate,
the precision of DC power model could meet the demand, though it is not so much accuracy as AC power model. In
the following, the static TNEP problem is formulated as a mixed integer nonlinear
programming model, and the power network is represented by a DC power flow model
………………………….3
………………………….4
………………………….5
………………………….6
…………………………..7
…………………………..8
where,
ci j is Cost of the addition of a circuit in branch i-j.
xi j is Number of new circuits added in branch i-j.
x0i j is Number of circuits in branch i-j in the initial case.
xi j is Maximum number of new circuits added
in branch i-j. Pi j is Power flow in branch i-j.
Pi j is Maximum power flow of a circuit in branch i-j.
Pgi is Generated power by units at bus i.
Pdi is Demand power of loads at bus i.
bi j is Susceptance of a circuit in branch i-j.
di is Voltage phase angle at bus i. W is Set of all candidate circuits.
Substitution in the above 6 for Pi j given by 5 yields
In the above programming model;
Eq. 3 is the objective function representing the investment costs of new transmission facilities.
Eq. 4 models the power network by Kirchoff’s Laws (KCL and KVL).
Eq. 5 is DC power flow equation.
Eq. 4 and 5 is the constraints that power flow must satisfy.
Eq. 6 is the constraint of maximal transmission power in the branch.
Eq. 7 is the constraint of the output power of a generator.
Eq. 8 is the maximal number of new circuits in a branch that can be added.
There are two unknown variables xi j and di multiplying in 5, so it’s a nonlinear constraint.
Considering variable xi j is integer, therefore, the whole is a hybrid nonlinear integer programming model.
Transmission System Operator (TSO).
Under this model, TSO is a single buyer who purchases
energy from GENCOs and sell it to DISCOs. Assume that
there is no direct contract between GENCO and DISCO. All
transactions must be cleared in the spot market. The social
welfare is calculated from the summation of participant
welfares subtract the expansion cost as shown in (1).
………………1
where
Here SW is the social welfare, GW1, CWj and TOW are the
welfares of i-th GENCO, j-th DISCO and TSO, respectively.
INV is the transmission expansion cost, N and M are the
numbers of GENCOs and DISCOs. GRVy, GBC, and GTUC,
are the revenue, bid (supply function) and transmission usage
charge of i-th GENCO. CUj, CTCj,CTUCj and CTUCj are the
bid (demand function), cost of energy, outage cost and
transmission usage charge of j-th DISCO. TORV and TOTC
are the revenue and cost of TSO, respectively.
After substituting (2)-(6) for (1), social welfare can be
rewritten as:
It should be noted that social welfare in case where
TRANSCO and TSO have been separated can be derived by using the
similar way. It also provides the same result.
where gi and Ij are the schedule outputs of i-th GENCO andjth
DISCO in the spot market. gimin, gimax, 'jmin, 'jmax are the
lower and upper limits of i-th GENCO and j-th DISCO,
respectively. LFq is the line flow in q-th transmission line,
LFqmax is the capacity of q-th transmission line, Q is the
number of transmission lines.
However, when there is unavailability of generation and/or
transmission facility, the redispatch process will be conducted
to maintain system security. The redispatch process is the
optimization problem which determines the amount of power
of generator to be changed and the amount of load to be
curtailed. Detail of redispatch can be found in Appendix.
In the redispatch process, constraint (9) must be modified
to (13). Moreover, the additional constraints in (14) and (15)
must be included.
13
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15
Here gi and Ij are the original schedule outputs of i-th
GENCO and j-th DISCO, Ag, and EENSj are the amount of
power of i-th GENCO to be changed and expected energy not
supplied of load ofj-th DISCO due to facility unavailability.
If there is only inelastic demand, DISCO demand function
can be neglected. Hence, the objective function is changed to
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It should be noted that the formulation is represented on
hour-based. This is because hour-based is the most basic
phase and it can be extended to cover a long-term
transmission expansion planning without difficulty.
III. METHODOLOGY
A. Assumption
In this paper, we have some assumptions as follows:
1. Only one transmission expansion project is considered.
2. The transmission expansion cost (INV) varies
proportionally to the capacity and length of the transmission
line.
3. The DC power flow method is employed and the spot
market is cleared by using the nodal price mechanism.
B. Uncertainty
The uncertainties which are considered are as follows:
1. Uncertainty in generation-load pattern
2. Uncertainty in bidding strategy of market participant
3. Availability of generation and transmission facilities
Generation investment and load growth can be predicted
based on many factors such as geography, resource and
government policy. Because usually, there are not so much
possible generation-load patterns (capacity and position), this
uncertainty is handled by a set of generation-load patterns
with their probabilities.
The bidding strategy of participant in the spot market is
modeled based on the real behavior in New England [4].
According to the historical data, GENCO bid can be divided
into two parts. In the first part, GENCO bids based on the real
marginal cost of his/her units. Usually, this strategy is applied
from 0 MW to about 80% of capacity. In the second part,
GENCO bids extremely high for the rest of capacity. Bidding
strategy can be showed as in Fig. 2. Slopes ml and m2 present
the strategy of the first and second parts, respectively. The
uncertainty in GENCO bidding strategy is handled by using
Monte Carlo simulation. For each scenario, ml and m2 are
generated randomly using Weibul and Normal distributions as
shown in (18) and (19), respectively.
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Availability of generation and transmission is handled by
using Monte Carlo simulation. These availabilities are
randomly generated based on their availability data.
C. Optimal Transmission Plan Determination
A combined approach between analytical method and
Monte Carlo simulation is proposed to solve transmission
expansion planning problem. The calculation procedure can
be described step by step as follows:
1. Start from the first generation-load pattern p = 1
2. Obtain generation, load and transmission line data
3. Set Monte Carlo simulation scenario k= 1
4. The bidding strategies of GENCO and DISCO are
randomly generated.
5. Spot market is cleared, and the nodal price (NPpk),
scheduled output of generation (gipk) and load (4jpk) are
determined.
6. Generation and transmission facilities' availabilities are
randomly generated.
7. If some of generation and/or transmission facilities are
unavailable, the redispatch process is applied in order to
determine the amount of power of generator to be changed (A
gipk) and the amount of load to be curtailed (EENSjpk).
8. Compute SWpk using equation (8)
9. Repeat from step 3 to step 8 until the predefined
number of scenarios (K) is reached. Then, compute SWp and
NPp from the average value of SWpk and NPpkof all scenarios.
10. Repeat step 1 until the number of generation-load
patterns (P) is reached. Social welfare and nodal price before
expansion (SWo and NPo) are computed by using (20) and (21).
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